You'll be in the mountains, and it gets chilly. I've done Camino del Norte twice in May and it was never hot, so I would say no to shorts, but convertible pants would be a better option because then you have both
You could also consider to additionally bring a skort because you will have warm days.
Lightweight synthetic/down jacket is a must and it won't take much room in your pack (I love my Mont-bell Superior Down Parka at 7.7 oz (217 g). Definitely bring a long sleeved top - preferably a lightweight wool, and have a short sleeve one too, in a lightweight wool. I'm partial to wool because of all its marvelous qualities - especially not retaining odor, and I can attest to that from years of experience
Also, of course a rain jacket and if you can, take a lightweight windbreaker, it comes in very handy
If there is anything you want to ask me, please PM me. Just hover over my picture and click on my name.
Otherwise, there are so many generous, knowledgeable and experienced members that are always willing to help. Don't hesitate to ask as many questions as you deem necessary to get to the point where you feel comfortable in your preparations for your camino.
Buen Camino!